Phase control as basis for precise impedance bridge microminiaturization
M. Surdu1, D. Surdu2
1
Institute of Electrodynamics, Kiev, Ukraine, msurdu@mbi.com.ua
2
Ukrmetrteststandard, Kiev, Ukraine, lanceoflife@gmail.com
Abstract
The new approach to the creation of the
equivalent voltage dividers is proposed. This
approach is based on the synthesis of the signal,
having digitally controlled magnitude by the algebraic
summing of the signals, having digitally controlled
phase. On this base one of the possible bridges
structure has been developed and its properties have
been analyzed
Key words: phase control, precise
impedance parameters measurements.
Introduction
For the precise impedance measurements on
audio frequency range in main laboratories usually
simple [1-13] or quadrature [14-21] transformer
bridges are used. Such bridges contain, as main part,
some (2-6 or more) precision transformer dividers. It
led to great dimensions of the devices, their narrow
frequency range and high cost. These disadvantages
sharply increase on the lower frequencies. It makes
impossible creation of the transformer bridge for low
frequencies. Therefore, replacing of the inductive
divider in impedance measurements by another
device could be very useful.
Last time the digital synthesis of the signals
and creation on this basis of different bridges are
widely used [22-25]. But these bridges can’t compete
in accuracy with transformer bridges.
So, the main today problem consist in the
following: most accurate transformer bridges have
great dimension and cost and really can’t operate on
low frequency range.
Solution
The main idea of this article is shown on Fig
.1. Here, on the plane of complex numbers (1; j.), are
shown basic signal (vector U0), which coincides with
real axis and two additional vectors U11 and U21.
Δψ 11
+ψ
−ψ
Δψ 21
The bisectrix, dividing onto two angle
between vectors U11 and U21, is turned relatively on
the real axis on angle φ.
Basic and additional signals (vectors U0, U11
and U21) could have different magnitudes and phases
but it is preferable to use the signals, satisfying to the
equality:
U 0 = U11 = U 21 ; ψ 11 = −ψ 21 .
We will sum additional vectors U11 and U21.
+
The total balancing vector U1 will lie on the
+
bisectrix. It is easy to show that this signal U1 is
described by equation:
U1+ = 2 U 0 cosψ sin(ωt + ϕ ) .
(2a)
Let us subtract additional vectors U11 and
U21. (the subtraction can be submitted by summing of
two signals one of them being negative). The
−
differential
balancing
vector
U 1 will be
perpendicular to the bisectrix. It is easy to show that
−
the signal U 1 is described by equation:
U 1− = 2 U 0 sinψ cos(ωt + ϕ ) .
(2b)
We will change angles ψ and –ψ
simultaneously from zero to the same current value
ψ
.
Equations (2a) and (2b) evidently show that
balancing signal U1 in both these cases could be
changed in magnitude and phase by controlling of
the phases ψ and φ only.
The range changing of the U1 magnitude in
both cases lie between 0 and ±2U0. The range
changing of its phase lies between 0 and ± 180. It is
obviously that using the signals U0 and U1 we could
create universal balanced bridges for measurements
of any type impedances using the changing of the
phase only.
The dependence of the vector U1 magnitude
on the phase angle ψ is nonlinear. It doesn’t
influence on the accuracy of the measurement
because of this dependence is strictly calculated. But
the slope of this dependence (see equations (2a) and
(2b)) change in the whole range of angle ψ from 0 to
1. The zero value of the slope could create
difficulties in automated bridge balancing. If we will
limit the range of the vector U1 changing, for
example, to U 1 ≤
Fig.1
(1)
2 U 0 the slope will change
from 0.5 to 1 only. Such change of the slope will not
influence the balancing process and only slightly
( 2 ) restricts range of measurement.
Described above system of vectors isn’t
unique, which permits to create the balancing vector,
controlled in magnitude by phase control. It is just the
simplest one only. The calibration procedure of the
bridges, using this system, is the simplest as well.
Signals U0, U11 and U21 could be easily
created using today digital technique. For this purpose
usually the different synthesis circuit (synthesizers),
based on the fast digital-to-analog converters, are
used [7, 8.].
Such synthesizers in audio frequency range
have rather great uncertainty - usually around 10-4 or
worse. But these synthesizers could have enough
good short term (during one measurement - one
minute or less) stability.
If the initial vectors magnitude ratios and the
phase sift differences from the nominal would be
determined and excluded, only its short term
instability and accuracy of the phase changing will
influence on the common uncertainty of the
measurement.
Using proper components, their temperature
stabilization, etc. synthesizers short term instability
could be reduced to values 10-8 or less.
Phase changing uncertainty is restricted by
components phase noise only. Today this value could
be easily reduced to 10-9 or less.
Of course, the components noise, caused by
different sources (low frequency noise, noise, caused
by charge injection in DAC, etc) will influence the
measurement resolution.
The lot of common problems, arising during
the impedance measurements and widely discussed
and solved in lot of excellent classic works, (such as
four pair terminal connections, etc) could restrict the
measurement uncertainty as well. But we will not go
here into all these very interesting and important
problems.
A lot of different bridges could be created
using described approach. Here we will show the
simplest bridge with current comparison and on this
example will discuss the peculiarities of its balancing
and problems of its calibration.
Fig.2 shows the functional diagram of the
possible bridge, using proposed idea. The bridge
consists of the stable DC voltage source U=,
connected to inputs of three synthesizers S11, S21 and
S0. These synthesizers create appropriate sinusoidal
signals U11, U21 and U0 under the microcontroller MC
control.
The output sinusoidal signals of the
synthesizers S11, and S21 are summed by adder Σ,
creating balancing signal U1. The impedances to be
compared Z0 and Zx are connected to the outputs of
synthesizer S0 and adder Σ. The vector voltmeter VV
measures the bridge output signals through switcher
C2. The microcontroller MC processes the results of
the vector voltmeter VV measurement and control all
the bridge balancing procedure by changing of the
appropriate signal phases.
Fig.2
Bridge balancing.
The balance condition of the bridge is
described by very simple equation:
r
Z x U1b
= r
Z 0 U ob
(3)
Let we’ll balance the bridge, changing the
magnitude and the phase φ of the vector U1. In this
case the equation could be rewritten in the form:
r
Z x U 1b
= r = (2 cosψ b )e − jϕb = ρ b e − jϕb (4)
Z 0 U ob
Equation (4) shows with evidence that we
can get direct reading, if we will measure the
magnitude and phase of the ratio
where:
Zx
= ρ x e − jϕ x ,
Z0
Zx
= ρx .
Z0
In this case the balancing equation (4)
divides onto two simplest:
ρ x = ρb = 2 cosψ b
and
ϕ x = ϕb .
(5)
where: ψb and φb – the coordinates of
balance point.
Of course, if we know these two parameters
of impedance ratio any other desired parameters of
this ratio, as well as the impedance Zx parameters,
can be easily calculated.
To balance the bridge we could use
variational method [10]. In this case we firstly
measure initial bridge unbalance signal Un1. After it
we provide the variation of the bridge parameters,
used for its balancing (the angles ψ or φ), and
measure new unbalance signal. For certainty, let we
will change the ψ, adding Δψv and measure the new
unbalance signal Un2. Next system of equations will
describe this process:
r
ρxe− jϕ U0 − U0 ρe− jϕ = Un1(Zx + Z0 ) / Z0
x
(6)
r
ρxe− jϕx U0 = U0 (ρ + Δρv )e− jϕ = Un2(Zx + Z0 ) / Z0
Here
ψ = ψ b + Δψ
and
ϕ = ϕb + Δϕ ,
where: Δψ and
Δφ is the distance between
coordinates of current bridge point (ψ, φ) and
coordinates (ψb, φb) of point of balance. Solving the
system (6) we find these distances from next implicit
function:
ρ x − jΔϕ
e
= (1 − Aδ v ) ;
ρ
(7)
where:
r
r
U n1
U n1
r = r
r e − jϕa = A e − jϕa ;
A= r
U n 2 − U n1 U n 2 − U n1
cos(ψ + Δψ v )
δv =
−1 .
cosψ
Calculating and entering in the bridge the
coordinates of balance point, we achieve the full
bridge balancing.
Bridge calibration
To calibrate the bridge let we will perform
balance equation. It is easy to see that additional
r
r
U11 and U 21 could be described by
r
r
r
r
equations: U 11 = U11n + ΔU11 = U11 (1 + δ11 ) and
r
r
r
r
U 21 = U 21n + ΔU 21 = U 21 (1 + δ 21 ) . These signals
r
r
have constant nominal values U11n and U 21n and
constant relative deviations from nominal δ11 and
δ 21 as well. Using these formulas we could rewrite
signals
U
= 1n + (δ 11 + δ 21 ).
U0
where:
δ 21 = ΔU 21 /U 0
δ11 = ΔU11 /U 0
2. At the second stage, the MC varies the
synthesizer S11 transfer coefficient on δ11v. After that,
r
thevoltmeter measures the unbalance signal U n 2 .
r
signal U n 3 .These measurements are described by
the following system of equations:
r
r
r
r U 0 − U 0 (1 + δ11 )
U0 −
⋅ Z1 − U n1 = 0,
Z1 + Z 2
r
r
r U 0 − U 0 (1 + δ11 + δ11v )
r
U0 −
⋅ Z1 − U n 2 = 0, (9)
Z1 + Z 2
r
r
r U − U 0 (1 + δ11 )
r
U0 − 0
⋅ Z 2 − U n 3 = 0.
Z1 + Z 2
(8a)
or:
r
r
r
r
Z x U 11n + U 21n ΔU 11 + ΔU 21
=
+
=
r
r
Z0
U0
U0
r
signal U n1 .
3. At the third stage, the MC reverses the
switcher C2 and the vector voltmeter measures the
balance equation into two equivalent forms:
r
r
r
r
Z x U 11 n + Δ U 11 U 21 n + Δ U 21
=
+
=
r
r
Z0
U0
U0
r
U 11 n
U
= r (1 + δ 11 ) + 21 n (1 + δ 21 ).
U0
U0
Equation forms (8a) and (8b) correspond to
two different calibration procedures. Both these
procedures are based on consistent nulling of one of
the components in formulas (8a), (8b) and
determination of the remains component.
First calibration procedure, based on
formula (8a), consists of two similar steps:
1. Calibration of the synthesizers S11.
2. Calibration of the synthesizers S21.
1. On the first step of the calibration
procedure we switch off the operation of the
synthesizer S21 (for example, entering and
maintaining zero control codes into the synthesizer
S21 or simply switching its output from the
appropriate adder input).After it we set into
synthesizer S11 the codes, corresponding to equality
U0= -U11n. In this case the outputs of the synthesizer
S0 and adder could be considered as input and output
of the appropriate inverter and the calibration
procedure, described in [9], could be used.
Calibration circuit, consisting of standards
Z1 and Z2, is connected to outputs of the synthesizer
S0 and adder Σ through switcher C1. Last one
reverses phase of the calibration circuit connection to
the mentioned signal sources during the calibration
process (fig.2).
All procedure is based on variation and
replacing methods [26, 27] and consists of three
stages.
1. At the first stage, the switcher C2 connects
the vector voltmeter to the output of the Z1-Z2
divider. Switcher C1 remains in the initial position
and the vector voltmeter measures the unbalance
. (8b)
and
- relative deviations of the
additional vectors U11 and U21 from nominal values
U11n and U21n. Values δ11 and δ 21 don’t depend on
the angles ψ and φ.
Neglecting the second order terms we will
get the next result:
r
r
r .
δ11 =
⋅ r
2 U n 2 − U n1
δ11v U n1 − U n 3
(10)
So, using the results of the three mentioned
measurements and equation (10), we can find the
synthesizer S11 relative deviation of the transfer
coefficient from nominal. The result of the
measurement of the deviation does not depend on
additive or multiplicative voltmeter errors. The
accuracy of the error synthesizers S11 determination
depends on the voltmeter nonlinearity and sensitivity
only.
2. On the second step of the calibration
procedure we switch off the operation of the
synthesizer S11. After it we set into synthesizer S21 the
codes, corresponding to equality U0= -U21n and repeat
described previously calibration procedure, including
next three measurements. In such way we get the
value δ21 = ΔU21/U0.
It is easy to show that to correct the result of
the ratio Zx/Z0 measurement, the U1 has to be divided
on 1+ δ11 + δ21.
Described calibration procedure doesn’t
needs using of accurate standards in calibration
circuit. It needs their short term stability during the
calibration process only. But it needs six
measurements, which could take rather long time.
Using of the relatively high resistive calibration
divider decrease the possible power of the signals to
be measured as well.
Another calibrating procedure, taking less
time and less number of measurements, is illustrated
by Fig.3. This procedure is based on second form of
balance equation (8b).
δ11 + δ 21 =
U n1
δv
U n2
(12)
As earlier, this result can be used for
uncertainty correction.
Uncertainty of the value δ11 + δ21 estimation
doesn’t depend on vector voltmeter gain error. It
depends, as earlier, on its nonlinearity and
sensitivity. In addition, it depend on vector voltmeter
zero shift. This one has to be measured and
eliminated from results of measurements using well
known procedures. In case of the second calibration
procedure correction uncertainty depends on additive
divider AD accuracy as well. For example, if the δ11
+ δ21= 10-4 and we want to get component of
common uncertainty, created by δ11 + δ21, being less
than 10-8, the uncertainty of AD transfer coefficient
δv has to be less than 10-4.
Preliminary experimental investigations of
the described bridge have shown that uncertainty of
the measurement in real condition, without
temperature stabilization of the used components,
achieve 10-7. These investigations have shown as
well, that bridge resolution depends on the noise,
caused by charge injection in used DAC. This noise
quickly increase then operation frequency increase.
On low and infralow frequency range the
contribution of this noise to common uncertainty
sharply decrease. But on this frequency range the
flicker noise increase its influence on the result of
measurement.
Conclusion
Fig.3.
Here the additional divider AD is connected
to the output of the synthesizer S0. This divider
transfers the synthesizer S0 output signal with transfer
coefficient δv. The AD and Adder outputs are
connected to the switch C1 inputs. Calibration
procedure consists of two measurements only.
1. During first step we set into synthesizers
S11 and S21 the codes, which correspond to equality
U11n= -U21n. In this case on the adder output will act
the signal ΔU11+ ΔU21. Through switchers C1 and C2
the vector voltmeter measures this signal as Un1.
2. On the second step the switcher C1 is
turned of, so that vector voltmeter measure the AD
output signal Un2. These two measurements are
described by simple system:
ΔU 11 + ΔU 21 = U n1
δ vU 0 = U n 2
(11)
It is easy to show that the total deflection
from nominal here will be determined by equation:
Two features of this bridge have to be
underlined:
Firstly, we have got possibility to measure
accurately impedances in wide and continuous range
of values without using of the inductive dividers.
And, second, we have got possibility to
measure impedances accurately in wide frequency
range.
In principle, there are no any limitations in
region of low frequency (the time of measurement
only).
In high frequency range there is the
limitation, caused by operation speed and number of
digits in synthesizers. Last one determines the
discreteness of bridge balancing only. Regard to the
fact that today digital-to-analog converters have the
operating frequency more than 10 MHz and
discreteness 10-3-10-5, we could create bridges with
highest operating frequency up to units of kHz.
Preliminary experimental investigations
have shown that theoretical ideas of this article could
be successfully realized and we could create on this
basis bridges, measuring impedance ratio with
uncertainty up to 10-7 and less.
Aknowlegments
The authors are very grateful to Dr. H.
Bachmair, Dr. M. Klonz (PTB, Germany), Dr. Y.
Wang , Dr. A. Koffman and Dr.J. Kinard (NIST,
USA) for their constant support and very useful
advice during the development of these ideas.
References
1. Kibble B.P., Rainer G.H., Coaxial
alternative current bridges., Briston; Pdam Hilder
Ltd.,1984.,203 p.
2. G. Small, Two- terminal - port AC bridges
at NML, BNM-LCIE, 1998. p.B1-B18
3. F. Delahaye, AC-bridges at BIMP, BNMLCIE. 1998. p.C1-C6.
4.
A.M.
Thompson,
The
Precise
Measurement of Small Capacitances, IRE Trans Instr.
1958.
5. A.-M. Jeffery, J. Shields, Scott Shields,
L.H. Lee, New Multi-frequency Bridge at NIST.
BNM-LCIE. 1998, p.G1-G37.
6. G. Eklund, Low frequency AC-bridges at
SP. BNM-LCIE. 1998. p.H1-H11
7. G. Trapon, Coaxial AC bridges at BNMLCIE. BNM-LCIE. 1998. p.I1-I22.
8. Awan S.A., Giblin S.P., Chua C.W.,
Hartland A., Kibble B.P. Summary of recent AC
QHR measurement at the National Phisical
Laboratory.BEMC-99- International Conference on
Electromagnetic Measurement, Brinton, UK, 32-4
November, 1999, 60/1-60/4
9. Трансформаторные измерительные
мосты., Под. ред. К.Б. Карандеева., Москва.,
«Энергия».,1970 г. 280 с
10. Avramov S.,Oldham N.M., Jarrett D.G.
and Waltrip B.C., Automatic Inductive Voltage
Divider Bridge for operation from 10 Hz to 100 kHz.,
Conf.Record.
of
Conference
on
Precision
Electromagnetic Measurements (CPEM’92), June., 912., 1992., Paris., France., pp.419-420.
11. Cutkosky R.D. An automatic high –
precision audio frequency capacitance bridge., IEEE
trans. Instr.Meas. vol IM-34., September., 1985.,
p383-389.
12. Anden-Hagerling., 1 kHz Automatic
Capacitance Bridge AN2500., Instruction Manual.
13. Jeffery A-M, Shields J.Q, Shields S.H. A
Multy-Frequency 4-Terminal – Pair AC Bridge.
CPEM 2000, Conferense Digest, pp 437-438
14. B. Wood, M. Cote, AC bridges for the RC Chain. BNM-LCIE. 1998. p.E1-E20.
15. S.W. Chua, B.P. Kibble, A. Hartland,
Comparison of Capacitance with AC Quantized
Hall Resistance. BNM-LCIE. 1998. p.G1G37.
16. Chua S.W., Kibble B.P., Hartland A.,
Comparison of capacitance with AC Quantized Hall
Resistance., 1998, Conference on Precision
Electromagnetic Measurements, Washington DC,
USA,6-10 Jily,pp.418-419
17. Гурьянов В.С. Метод получения квадратурных напряжений с равными амплитудами.,
Метрология.,1986г. № 4., c. 54.
18. Гурьянов В.С. Сурду М.Н., Салюк
В.П., Универсальные трансформаторные мосты
переменного тока для передачи размеров единиц
параметров комплексного сопротивления. «Техническая электродинамика», 1991., № 3., с. 103108.
19. Trapon G, Thevenot O, Lacueille J.C,
Poirier W, Fhimia H, Geneves G. Progress in linking
of the Farad and the Rk to the SI Units at
BNM_LCIE. CPEM 2000, Conference Digest,
pp214-215.
36.Thompson
A.M.,
An
absolute
determination of resistance based on calculable
standard of capacitance., Metrologia, Vol. 4, № 1б
pp 1 – 7, January, 1969.
20. Hsu J. C, Yi-sha Ku. Comparison of
capacitance with resistance by IDV-based
Quadrature bridge at frequencies from 50 Hz to 10
kHz. CPEM 2000. Conference Digest. pp. 429-430.
21. Nakamura Y, Nakanisi M, Sacamoto Y,
Endo T. Development and uncertainty estimation of
bridges for the link between capacitance and QHR at
1 kHz. CPEM 2000, Conferense Digest, pp 430-431.
22. Cabiati F.,Bosco G.C. L-C comparison
system based on two-phase generator. IEEE Trans.
Instrum. Meas. 1985., 34., N.2., p. 344-349.
23. Corney A.C. Digital Generator Assisted
Impedance Bridge CPEM 2002, Conference Digest,
pp176-177.
24. Callegaro l. D’Elia, Bava E., Galzerano,
Svelto C., Polyphase Synthesizer for UnlikeImpedance Intercomparison System. CPEM 2002.
Conference Digest. pp.176-177.
25. Cabiati F, D’Elia. A New Architecture
for High Accuracy Admittance measuring system.
CPEM 2002, Conference Digest. Pp.178-179.
26. M.N.Surdu, A.L. Lameko, I.V. Karpov,
J.Kinard, A. Koffman. Theoretical basis of
variational quadrature bridge design on alternative
current. Moskow, Measurement Techniques, № 10,
2006 y. pp 58-64.
27. Гриневич Ф.В., Сурду М.Н. Высокоточные вариационные измерительные системы
переменного тока., Киев., Наукова Думка., 1989.,
192 с.